Quantitative susceptibility mapping image processing method using neural network based on unsupervised learning and apparatus therefor

ABSTRACT

Disclosed is a quantitative susceptibility mapping image processing method using an unsupervised learning-based neural network and an apparatus therefor. The quantitative susceptibility mapping image processing method includes receiving a phase image and a magnitude image for reconstructing the quantitative susceptibility mapping image, and reconstructing the quantitative susceptibility mapping image corresponding to the received phase image and the received magnitude image using an unsupervised learning-based neural network, and the neural network may be generated based on an optimal transport theory.

CROSS-REFERENCE TO RELATED APPLICATIONS

A claim for priority under 35 U.S.C. § 119 is made to Korean PatentApplication No. 10-2021-0043432 filed on Apr. 2, 2021 and No.10-2022-0017320 filed on Feb. 10, 2022, in the Korean IntellectualProperty Office, the entire contents of which are hereby incorporated byreference.

BACKGROUND

Embodiments of the inventive concept described herein relate to aquantitative susceptibility mapping image processing technique using anunsupervised learning-based neural network, and more particularly, to aquantitative susceptibility mapping image processing method andapparatus capable of reconstructing the quantitative susceptibilitymapping image using the unsupervised learning-based neural network.

Quantitative susceptibility mapping (QSM) is a magnetic resonanceimaging (MRI) technique that can quantitatively measure the degree ofsusceptibility of body tissues in a magnetic field. QSM may providespecific contrast images or provide information that is helpful indiagnosing diseases because QSM is sensitive to biomarkers such as irondeposition.

A quantitative susceptibility mapping image may be obtained from a phaseimage of a magnetic resonance image. From a mathematical point of view,the phase image of the magnetic resonance image is expressed by theconvolution between the quantitative susceptibility mapping image and adipole kernel. A quantitative susceptibility mapping image may beobtained by dividing a phase image by a dipole kernel in Fourier space,and this process is called dipole inversion. Thus, a magneticsusceptibility distribution can be obtained by deconvolution: forexample, dividing the phase by the dipole kernel in the Fourier domain.However, the dipole kernel in the Fourier domain has zeros along theconical surface, which makes the dipole inversion an ill-posed inverseproblem.

To overcome these limitations, various dipole inversion approaches havebeen investigated. Calculation of susceptibility through multipleorientation sampling (COSMOS) is considered as the golden standardalgorithm for the dipole inversion. COSMOS restores accurate QSM frommultiple head orientation data. However, the acquisition of MR dataalong multiple head orientations for COSMOS reconstruction takes toomuch subject's time and effort.

Accordingly, algorithms have been developed for dipole inversion fromsingle head orientation data. For example, filling the area around theconical surface with a threshold value is one of the methods for dipoleinversion. Also, the regularization by edge information of the magnitudeimage can mitigate the ill-posedness of dipole inversion. In addition,dipole inversion algorithms based on compressed sensing have beenstudied. However, the streaking artifact in the reconstructed QSM andthe difficulty of hyperparameter tuning are limitations of the abovealgorithms.

Recently, deep learning algorithms have also been extensively studiedfor QSM reconstruction. These algorithms show comparable performance asthe classical methods despite the fast computational time. However, mostof the existing deep learning methods for QSM reconstruction are basedon supervised learning which requires matched pairs of phase images andground-truth QSM labels. Nonetheless, it has been reported that thereconstructed QSM values are often underestimated.

SUMMARY

Embodiments of the inventive concept provide a quantitativesusceptibility mapping image processing method capable of reconstructinga quantitative susceptibility mapping image using an unsupervisedlearning-based neural network and an apparatus therefor.

According to an exemplary embodiment, a quantitative susceptibilitymapping image processing method for reconstructing a quantitativesusceptibility mapping (QSM) image includes receiving a phase image anda magnitude image for reconstructing the quantitative susceptibilitymapping image, and reconstructing a quantitative susceptibility mappingimage corresponding to the received phase image and the receivedmagnitude image using an unsupervised learning-based neural network,wherein the neural network includes a cycle-consistency generativeadversarial network (cycleGAN) structure including at least onegenerator and at least one discriminator to reconstruct the quantitativesusceptibility mapping image.

The neural network may be generated based on an optimal transporttheory.

The neural network may be trained using a training dataset containingnon-matching data.

The neural network may be subjected to unsupervised learning by using afirst neural network that generates a first quantitative susceptibilitymapping image corresponding to a first phase image and a first magnitudeimage after receiving the first phase image and the first magnitudeimage, a first transform unit that performs Fourier transform on thefirst quantitative susceptibility mapping image, multiplies the firstquantitative susceptibility mapping image by a dipole kernelcorresponding to the first phase image and the first magnitude image,and performs inverse Fourier transform on the first quantitativesusceptibility mapping image to generate a second phase image, a secondtransform unit that performs Fourier transform on a second quantitativesusceptibility mapping image, which is a ground-truth image, andmultiplies the second quantitative susceptibility mapping image by adipole kernel corresponding to the second quantitative susceptibilitymapping image, and performs inverse Fourier transform on the secondquantitative susceptibility mapping image to generate a third phaseimage, a second neural network that generates a third quantitativesusceptibility mapping image corresponding to the second phase image anda second magnitude image after receiving the second phase image and thesecond magnitude image, and a third neural network that discriminatesthe first quantitative susceptibility mapping image and the secondquantitative susceptibility mapping image. Here, the first neuralnetwork and the second neural network may be the same neural network.

Furthermore, the neural network may be subjected to the unsupervisedlearning based on a cycle-consistency loss and a gradient differenceloss calculated by comparing the first phase image with the second phaseimage, or the second quantitative susceptibility mapping image with thethird quantitative susceptibility mapping image and a total variationloss for the first quantitative susceptibility mapping image, and anadversarial loss between the first quantitative susceptibility mappingimage and the second quantitative susceptibility mapping image.

The third neural network may distinguish a quantitative susceptibilitymapping image obtained by multiplying the first quantitativesusceptibility mapping image by a preset mask and the secondquantitative susceptibility mapping image.

The neural network may include any one of a neural network based on aconvolution framelet and a neural network including a pooling layer andan unpooling layer.

According to an exemplary embodiment, a quantitative susceptibilitymapping image processing method includes receiving a phase image and amagnitude image for reconstructing the quantitative susceptibilitymapping image, and reconstructing the quantitative susceptibilitymapping image corresponding to the received phase image and the receivedmagnitude image using an unsupervised learning-based neural networkgenerated based on an optimal transport theory, wherein the neuralnetwork includes a cycle-consistency generative adversarial network(cycleGAN) structure including at least one generator and at least onediscriminator to reconstruct the quantitative susceptibility mappingimage.

According to an exemplary embodiment, a quantitative susceptibilitymapping image processing apparatus includes a receiver that receives aphase image and a magnitude image for reconstructing the quantitativesusceptibility mapping image and a reconstruction unit that reconstructsthe quantitative susceptibility mapping image corresponding to thereceived phase image and the received magnitude image using anunsupervised learning-based neural network, wherein the neural networkincludes a cycle-consistency generative adversarial network (cycleGAN)structure including at least one generator and at least onediscriminator to reconstruct the quantitative susceptibility mappingimage.

The neural network may be generated based on an optimal transporttheory.

The neural network may be trained using a training dataset containingnon-matching data.

The neural network may be subjected to unsupervised learning by using afirst neural network that generates a first quantitative susceptibilitymapping image corresponding to a first phase image and a first magnitudeimage after receiving the first phase image and the first magnitudeimage, a first transform unit that performs Fourier transform on thefirst quantitative susceptibility mapping image, multiplies the firstquantitative susceptibility mapping image by a dipole kernelcorresponding to the first phase image and the first magnitude image,and performs inverse Fourier transform on the first quantitativesusceptibility mapping image to generate a second phase image, a secondtransform unit that performs Fourier transform on a second quantitativesusceptibility mapping image, which is a ground-truth image, andmultiplies the second quantitative susceptibility mapping image by adipole kernel corresponding to the second quantitative susceptibilitymapping image, and performs inverse Fourier transform on the secondquantitative susceptibility mapping image to generate a third phaseimage, a second neural network that generates a third quantitativesusceptibility mapping image corresponding to the second phase image anda second magnitude image after receiving the second phase image and thesecond magnitude image, and a third neural network that discriminatesthe first quantitative susceptibility mapping image and the secondquantitative susceptibility mapping image.

Furthermore, the neural network may be subjected to the unsupervisedlearning based on a cycle-consistency loss and a gradient differenceloss calculated by comparing the first phase image with the second phaseimage, or the second quantitative susceptibility mapping image with thethird quantitative susceptibility mapping image and a total variationloss for the first quantitative susceptibility mapping image, and anadversarial loss between the first quantitative susceptibility mappingimage and the second quantitative susceptibility mapping image.

The third neural network may distinguish a quantitative susceptibilitymapping image obtained by multiplying the first quantitativesusceptibility mapping image by a preset mask and the secondquantitative susceptibility mapping image.

The neural network may include any one of a neural network based on aconvolution framelet and a neural network including a pooling layer andan unpooling layer.

BRIEF DESCRIPTION OF THE FIGURES

The above and other objects and features will become apparent from thefollowing description with reference to the following figures, whereinlike reference numerals refer to like parts throughout the variousfigures unless otherwise specified, and wherein:

FIG. 1 is a flowchart illustrating an operation of a quantitativesusceptibility mapping image processing method according to anembodiment of the inventive concept;

FIG. 2 shows an exemplary diagram for an architecture of cycleQSM of theinventive concept;

FIGS. 3A and 3B show exemplary diagrams for a generator architecture anda discriminator architecture;

FIG. 4 is a diagram illustrating an example of a result ofreconstructing a quantitative susceptibility mapping image using amethod of the inventive concept and various conventional methods;

FIGS. 5A and 5B show diagrams illustrating an example for describing ananalysis result using susceptibility values of a ground-truth image andsusceptibility values of an image reconstructed by variousreconstruction methods; and

FIG. 6 is a diagram illustrating a configuration of a quantitativesusceptibility mapping image processing apparatus according to anembodiment of the inventive concept.

DETAILED DESCRIPTION

Advantages and features of the inventive concept and methods forachieving them will be apparent with reference to embodiments describedbelow in detail in conjunction with the accompanying drawings. However,the inventive concept is not limited to the embodiments disclosed below,but can be implemented in various forms, and these embodiments are tomake the disclosure of the inventive concept complete, and are providedso that this disclosure will be thorough and complete and will fullyconvey the scope of the invention to those of ordinary skill in the art,which is to be defined only by the scope of the claims.

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting of the inventiveconcept. The singular expressions include plural expressions unless thecontext clearly dictates otherwise. In this specification, the terms“comprises” and/or “comprising” are intended to specify the presence ofstated features, integers, steps, operations, elements, parts orcombinations thereof, but do not preclude the presence or addition ofsteps, operations, elements, parts, or combinations thereof.

Unless defined otherwise, all terms (including technical and scientificterms) used herein have the same meanings as commonly understood by oneof ordinary skill in the art to which this invention belongs. Further,unless explicitly defined to the contrary, the terms defined in agenerally-used dictionary are not ideally or excessively interpreted.

Hereinafter, preferred embodiments of the inventive concept will bedescribed in detail with reference to the accompanying drawings. Thesame reference numerals are used for the same components in thedrawings, and duplicate descriptions of the same components are omitted.

Embodiments of the inventive concept make it a gist of reconstructing aquantitative susceptibility mapping image using an unsupervisedlearning-based neural network generated by an optimal transport theory.

The neural network used in the inventive concept may include aconvolution framelet-based neural network, a neural network including apooling layer and an unpooling layer, for example, U-Net, as well asvarious types of neural networks applicable to the inventive concept.

A convolutional framelet refers to a method of representing an inputsignal through local and non-local bases. In order to reveal the blackbox characteristics of deep convolutional neural networks, the deepconvolutional neural networks are described in detail in a study on anew mathematical theory of deep convolutional framelets (Ye, JC., Han,Y., Cha, E.: Deep convolutional framelets: a general deep learningframework for inverse problems. SIAM Journal on Imaging Sciences 11(2),991-1048(2018)).

The contribution of the inventive concept is as follows.

Unlike the supervised learning approaches, the method of the inventiveconcept does not require matched QSM labels for the training so that thetrained model is less sensitive to the lack of training data with enoughvariability and structures. Accordingly, the underestimation issue ofQSM values in the supervised learning can be largely overcome.

In contrast to the existing unsupervised learning approaches, the methodof the inventive concept learns statistical properties of unpaired QSMlabels during training, which makes the training more stable, thusleading to the more accurate QSM map estimation with less outliers.

Unlike the classical approaches and deep image prior (DIP), the methodof the inventive concept is a feed-forward neural network that providesinstantaneous reconstruction once the cycleGAN training is done, thusmaking the algorithm very practical.

Prior to describing the inventive concept, conventional technology willbe briefly described below.

Various methods have been developed to solve the ill-posed dipoleinversion problem. For example, thresholded k-space division (TKD) is amethod of filling values near the conical surface of the dipole kernelwith a threshold value. Instead of filling the dipole kernel with aspecific value, some methods have used additional information for thedipole inversion. Another exemplary method has proposed homogeneityenabled incremental dipole inversion (HEIDI) using homogeneityinformation from gradient echo phase images. Morphology enabled dipoleinversion (MEDI) is a method that exploits the structural consistencybetween the magnitude image and the reconstructed quantitativesusceptibility map. Still another exemplary method has introduced amethod based on a sparse linear equation and least squares algorithm(iLSQR) to remove streaking artifacts in reconstructed QSM, andcompressed sensing based algorithms were also examined. Annihilatingfilter-based low-rank Hankel matrix approach (ALOHA) for QSM exploitsthe sparsity of the data in a certain transform domain to interpolatethe missing k-space data in the dipole spectral null. Although abovealgorithms showed high performance, there are limitations of the abovealgorithms, such as streaking artifacts, difficulties in optimizinghyper-parameters, and high computational complexity.

To overcome the limitations of traditional methods, QSM reconstructionalgorithms based on deep learning have been investigated. Conventionalexemplary methods have proposed QSMnet and deepQSM, respectively, whichare 3D U-net structures designed for QSM reconstruction. By using ofCOSMOS images as ground-truth QSM labels, the above methods showedcomparable results to those of classical approaches. Anotherconventional exemplary method has suggested QSMGAN where the adversarialloss was utilized, and still another conventional exemplary method hasintroduced xQSM where the octave convolutional layers were applied. Inaddition, still another conventional exemplary method has introducednonlinear dipole inversion with deep learning using a variational neuralnetwork, which combines optimization of nonlinear QSM data model and thedata fidelity term.

Although these methods have shown improved image quality compared toconventional methods, the requirement of matched pairs containing phaseimages and susceptibility maps is a limitation of aforementionedsupervised methods. Moreover, one of the biggest issues of supervisedlearning approach is the generalization error that happens when testdata have different characteristics to training data. For example,compared with the conventional QSM, deep learning QSM results may beunderestimated the susceptibility values when the susceptibility rangein the training data differs from the test data. Moreover, this effectis shown severe when training with synthetic data, which lacks enoughvariability and structure.

Recently, some deep learning algorithms based on weakly supervised orunsupervised learning were introduced. Conventional exemplary method hasproposed weakly supervised learning for QSM reconstruction (wTFI), andwTFI reconstructs QSM without background field removal, and enables therestoration of susceptibility values near the edges of the brain, whichcan disappear with background field removal. The conventional exemplarymethod has proposed unsupervised QSM reconstruction method (uQSM) bytaking advantage of nonlinear data consistency loss and the totalvariation loss.

FIG. 1 is a flowchart illustrating an operation of a quantitativesusceptibility mapping image processing method according to anembodiment of the inventive concept.

Referring to FIG. 1, a quantitative susceptibility mapping imageprocessing method according to an embodiment of the inventive conceptreceives a phase image and a magnitude image for reconstructing aquantitative susceptibility mapping image (S110).

When the phase image and the magnitude image are received in step S110,a quantitative susceptibility mapping image corresponding to thereceived phase image and magnitude image is reconstructed by using aunsupervised learning-based neural network (S120).

Here, the neural network used in the inventive concept may have a cycleGenerative Adversarial Network (cycleGAN) structure including at leastone generator and at least one discriminator to reconstruct aquantitative susceptibility mapping image.

In addition, the neural network used in the inventive concept may begenerated based on an optimal transport theory, and may be learned usinga training dataset including non-matching data.

Furthermore, the neural network used in the inventive concept may besubjected to unsupervised learning by using a first neural network(e.g., a generator) that generates a first quantitative susceptibilitymapping image corresponding to a first phase image and a first magnitudeimage after receiving the first phase image and the first magnitudeimage, a first transform unit that performs Fourier transform on thefirst quantitative susceptibility mapping image, multiplies the firstquantitative susceptibility mapping image by a dipole kernelcorresponding to the first phase image and the first magnitude image,and performs inverse Fourier transform on the first quantitativesusceptibility mapping image to generate a second phase image, a secondtransform unit that performs Fourier transform on a second quantitativesusceptibility mapping image, which is a ground-truth image, andmultiplies the second quantitative susceptibility mapping image by adipole kernel corresponding to the second quantitative susceptibilitymapping image, and performs inverse Fourier transform on the secondquantitative susceptibility mapping image to generate a third phaseimage, a second neural network (e.g., a generator) that generates athird quantitative susceptibility mapping image corresponding to thesecond phase image and a second magnitude image after receiving thesecond phase image and the second magnitude image, and a third neuralnetwork (e.g., a discriminator) that discriminates the firstquantitative susceptibility mapping image and the second quantitativesusceptibility mapping image. Here, the first neural network and thesecond neural network may be the same neural network.

Furthermore, the neural network may be subjected to the unsupervisedlearning based on a cycle-consistency loss and a gradient differenceloss calculated by comparing the first phase image with the second phaseimage, or the second quantitative susceptibility mapping image with thethird quantitative susceptibility mapping image and a total variationloss for the first quantitative susceptibility mapping image, and anadversarial loss between the first quantitative susceptibility mappingimage and the second quantitative susceptibility mapping image.

The third neural network may distinguish a quantitative susceptibilitymapping image obtained by multiplying the first quantitativesusceptibility mapping image by a preset mask and the secondquantitative susceptibility mapping image.

The neural network used in the inventive concept may include any one ofa neural network based on a convolution framelet and a neural networkincluding a pooling layer and an unpooling layer.

The method of the inventive concept will be described with reference toFIGS. 2 to 5.

Dipole Inversion

When the tissue is brought into the magnetic field, the tissue becomesmagnetized. Magnetic susceptibility “X” is the quantitative measure ofthe degree of magnetization. In MRI, the magnetization of tissuesgenerates the magnetic perturbation along the main magnetic field. Thismagnetic perturbation, or the phase signal can be represented as[Equation 1]:

b({right arrow over (r)})=d({right arrow over (r)})*X({right arrow over(r)}),r∈

³  [Equation 1]

where b is the phase signal, d is the dipole kernel, and * is theconvolution operation. Here, the dipole kernel d is represented by thefollowing Equation:

${d\left( \overset{\rightarrow}{r} \right)} = {\frac{1}{4\pi}\frac{{3\cos^{2}\theta} - 1}{{❘\overset{\rightarrow}{r}❘}^{3}}}$

where θ is the angle between {right arrow over (r)} and the mainmagnetic field, whose Fourier domain spectrum is given by [Equation 2]:

$\begin{matrix}{{\hat{d}\left( \overset{\rightarrow}{k} \right)} = {\frac{1}{3} - \frac{k_{z}^{2}}{{❘\overset{\rightarrow}{k}❘}^{2}}}} & \left\lbrack {{Equation}2} \right\rbrack\end{matrix}$

where {right arrow over (k)}=[k_(x), k_(y), k_(z)]^(T) is a k-spacevector.

Accordingly, one of the simplest dipole inversions may be achieved byelement-wise division in the Fourier domain, and may be expressed as in[Equation 3] below.

$\begin{matrix}{{\hat{\chi}\left( \overset{\rightarrow}{k} \right)} = \frac{\hat{b}\left( \overset{\rightarrow}{k} \right)}{\hat{d}\left( \overset{\rightarrow}{k} \right)}} & \left. \left( {{Equation}3} \right. \right\rbrack\end{matrix}$

However, [Equation 3] is not stable because {circumflex over (D)}({rightarrow over (k)}) has zero values at the conical surfac

$\left( {\overset{\rightarrow 2}{k} = {3k_{z}^{2}}} \right).$

Optimal Transport Driven CycleGAN

In the recent mathematical theory of optimal transport driven cycleGAN(OT-cycleGAN), it is revealed that various forms of cycleGANarchitecture can be obtained from the dual formulation of an optimaltransport problem, in which the transport cost is the sum of thedistances in the measurement and image domains. In particular, if theforward mapping is known from the imaging physics as in QSM, theresulting OT-cycleGAN architecture can be significantly simplified. Inthe inventive concept, the theory is applied for the QSM reconstructionand a measurement model as shown in [Equation 4] below is considered.

[Equation 4]

=

x

where y ∈

and x ∈X denote the measurement image and the unknown image,respectively, and

:X

is the known deterministic imaging operator.

In contrast to the supervised learning where the goal is to learn therelationship between the image x and measurement y pairs, there are nomatched image-measurement pairs in the unsupervised learning framework.Since sets of images and unpaired measurements can be still acquired, itis a goal to match the probability distributions rather than eachindividual samples. This can be done by finding transportation maps thattransport the probability measures between the two spaces.

Specifically, it is supposed that the target image space X is equippedwith a probability measure μ, whereas the measurement space

is with a probability measure ν. Then, it can be seen that the masstransport from (X,μ) to (

, ν) is performed by the forward operator H, and the mass transportationfrom the measurement space (

,ν) to the image space (x,μ) is done by a generator Θ, parameterized byG_(Θ):Y

X. Then, the following transportation cost is proposed for the optimaltransport problem as shown in the following [Equation 5].

c(x,y;Θ):=∥

−

x∥+G _(Θ)(

)−x∥  [Equation 5]

The [Equation 5] denotes the sum of the distance between a trainingsample and a transported sample in each space. Rather than minimizingthe sample-wise cost using [Equation 5], the goal of the optimaltransport is to minimize the average transport cost. More specifically,the optimal transport problem is formulated to find the jointdistribution π that leads to the minimum average transport cost, whichcan be expressed as in the following [Equation 6].

$\begin{matrix}{\inf\limits_{\pi \in {\prod{({\mu,v})}}}{\int_{x \times y}{{c\left( {x,{y;\Theta}} \right)}d{\pi\left( {x,y} \right)}}}} & \left\lbrack {{Equation}6} \right\rbrack\end{matrix}$

where Π(μ, ν) is the set of joint measures whose marginal distributionsin X and

are μ and V, respectively.

The geometric meaning of the optimal transport using [Equation 5] isexplained. More specifically, if the first term in [Equation 5] is onlyused, the optimal transport is to find the joint probability thatminimizes the distance in the empirical distribution V and the so-called“push-forward measure”^(ν)H. Similarly, if the second term in [Equation5] is only used, the optimal transport problem is to minimize thedistance in the empirical distribution μ and the push-forward measureμG. By using both terms in [Equation 5], the optimal transportformulation finds the joint measure that minimizes the sum of the twodistances in the measurement and the image spaces.

Using the transportation cost in [Equation 5], the Kantorovich dualformulation is given by [Equation 7] and [Equation 8],

$\begin{matrix}{{\underset{\Theta}{\min}{{\mathbb{K}}\left( {\Theta,\mathcal{H}} \right)}} = {\min\limits_{\Theta}\max\limits_{\Phi}{\ell\left( {\Theta;\Phi} \right)}}} & \left\lbrack {{Equation}7} \right\rbrack\end{matrix}$ $\begin{matrix}{{\ell\left( {\Theta;\Phi} \right)} = {{{\gamma\ell}_{cycle}(\Theta)} + {\ell_{WGAN}\left( {\Theta;\Phi} \right)}}} & \left\lbrack {{Equation}8} \right\rbrack\end{matrix}$

where γ is a suitable hyper-parameter, l_(cycle) is thecycle-consistency loss, l_(WGAN) is the Wasserstein GAN loss. Morespecifically, l_(cycle) and l_(WGAN) can be expressed by [Equation 9]and [Equation 10].

_(cycle)(Θ)=∫_(x) ∥x−G _(Θ)(

x)∥dμ(x)+∫_(y) ∥y−

G _(Θ)(y))∥dv(y)  [Equation 9]

_(WGAN)(Θ;Φ)=∫_(x)φΦ(x)dμ(x)−∫_(y)φΦ(G _(Θ)(y))dv(y)   [Equation 10]

Since the forward operator

is assumed known, it is noted that there exists only a singlediscriminator φΦ, so that there is no need to compete with the forwardoperator, making the cycleGAN architecture simple.

CycleQSM

-   In the dipole inversion, a phase image b ∈B and a susceptibility map    _(x) ∈X correspond to a noisy measurement and an unobserved image,    respectively. Therefore, the forward model of the dipole inversion    can be formulated as the following [Equation 11].

b=

⁻¹ {circumflex over (d)}

X  [Equation 11]

where

and

⁻¹ are 3D Fourier transform and 3D inverse Fourier transform,respectively.

By identifying and y:=b,x:=X and

:=F⁻¹ dF, the cycle-consistency loss and the GAN loss can be representedby the following [Equation 12] and [Equation 13].

_(cycle)(Θ)=∫_(x) ∥x−G _(Θ)(

x)∥dμ(x)+∫_(y) ∥y−

G _(Θ)(y))∥dv(y)  [Equation 12]

_(WGAN)(Θ;Φ)=∫_(x)φΦ(x)dμ(x)−∫_(y)φΦ(G _(Θ)(y))dv(y)  [Equation 13]

Although these costs can be used directly for the QSM reconstruction,the inventive concept provides additional modifications for a betterperformance. First, the inventive concept employs the least squares GAN(LSGAN) instead of WGAN loss for faster and more stable training. Thelink between WGAN and LSGAN was also explained in an existing study (S.Lim, H. Park, S.-E. Lee, S. Chang, B. Sim, and J. C. Ye, “CycleGAN witha blur kernel for deconvolution microscopy: Optimal transport geometry,”IEEE Transactions on Computational Imaging, vol. 6, pp. 1127-1138,2020.) Next, the gradient difference loss (l_(grad)) may be added tomaintain the edge information, and the gradient difference loss can beexpressed as in [Equation 14] below.

_(grad)(Θ)=∫_(x) ∥∇X−∇G _(Θ)(

⁻¹ d

X)∥dμ(X) +∫_(B) ∥∇B−∇

⁻¹ d

G _(Θ)(b)∥dν(b)  Equation 14

where ∇ is a gradient operator for the 3D volume.

To preserve image details and remove noise in reconstructed QSM, thetotal variation (TV) loss (l_(TV)) may be employed and the totalvariation loss may be expressed in [Equation 15] below.

TV(Θ)=∫_(B) ∥∇G _(Θ)(b)∥dν(b)  Equation 15

Therefore, the final cost function for CycleQSM can be formulated as[Equation 16] below.

(Θ;Φ)=γ

_(cycle)(Θ)+

_(LSGAN)(Θ,Φ)+η

_(grad)(Θ)+

TV(Θ)  Equation 16

where γ, η and ρ are appropriate hyper-parameters.

It may be noted that the losses in [Equation 14] and [Equation 15] areaverage values with respect to the marginal distributions. Therefore,the addition of these terms does not change the optimal transportinterpretation. In fact, [Equation 16] can be obtained as a dualformulation of the optimal transport problem with the followingtransportation cost:

c_(QSM)(b, χ; Θ) := b − F⁻¹d̂Fχ + G_(Θ)(b) − χ + η(∇χ − ∇G_(Θ)(ℱ⁻¹d̂ℱχ) + ∇b − ∇ℱ⁻¹d̂ℱG_(Θ)(b)) + ρ∇G_(Θ)(b)

where b ∈

and X^(∈X) with probability measures ν and μ, respectively.

To train and test algorithms, it is possible to use data sets fromdifferent sources. First data set is in vivo human brain data that wasprovided for 2016 QSM challenge. This data set may be acquired from ahealthy volunteer by 3T Siemens scanner with 3D gradient echo (GRE)sequence. Acquisition parameters of 2016 QSM challenge data are asfollows: resolution=1.06×1.06×1.06 mm³, matrix size=160×160, TR/TE=25/35ms. Also, this data set also contains ground-truth QSM which is acquiredby COSMOS with 12 different orientation data.

Next, the inventive concept may use in vivo healthy human data from theCornell MRI Research Lab. This data set may be collected using multiecho GRE sequence with 3T GE system. The acquisition parameters are asfollows: Resolution=0.9375×0.9375×1 mm³, matrix size=256×256×256,TR/TE₁=55/5 ms, and ΔTE=5 ms. QSM is also obtained by COSMOS with 5different head orientation data.

The third data set was obtained from five healthy volunteers using 3TSiemens system. This data set may be used for ALOHAQSM, and acquiredwith the following parameters: Resolution=0.75×0.75×1.l mm³, matrixsize=240×320×128, TR/TE₁=43/9.35 ms, and ΔTE=8.94 ms.

Last, the inventive concept may have used the data for 2019 QSMchallenge. This data set may be obtained through forward simulationbased on in vivo human brain data acquired on 7T MR system. Theacquisition parameters for 2019 challenge data are as follows:Resolution=0.64×0.64×0.64 mm³, matrix size=164×205, TR/TE₁=50/4 ms, andΔTE=8 ms. The local field data in 2019 challenge data set were simulatedwith two different contrast levels and two signal-to-noise ratio (SNR)levels, so total four local field maps (Sim1Snr1, Sim1Snr2, Sim2Snr1,Sim2Snr2) may be provided and, in addition, ground-truth QSM data wereprovided for two different contrast levels.

Table I shows the name and number of volumes of each data set. The 2016QSM challenge and 2019 QSM challenge data in Table I are used to trainour network as well as other deep learning approaches. For unsupervisedlearning, a matched reference data is not required. Therefore, threelocal field maps of the ALOHA-QSM data set are also used for trainingand the remaining two local field maps are used for inference, andfurthermore, the Cornell data set is used for the quantitativeevaluation of our algorithm.

TABLE 1 2016 QSM Challenge Cornell ALOHA-QSM 2019 QSM ChallengeResolution Voxel pitch (mm³) 1.06 × 1.06 × 1.06 0.9375 × 0.9375 × 1 0.75× 0.75 × 1.1 0.64 × 0.64 × 0.64 Matrix size 160 × 160 × 160 256 × 256 ×146 240 × 320 × 128 164 × 205 × 205 No. of Total 1 1 5 4 (2 SNR/2Contrast) Cases QSM Ground-truth 1 (COSMOS) 1 (COSMOS) 0 2 (Si

tion) Training 1 0 3 4 Test 0 1 2 0

indicates data missing or illegible when filed

For data preprocessing, brain mask extraction using the brain extractiontool (BET), multi-coil phase combination by Hermitian inner product(HiP), and multi-echo phase correction by nonlinear frequency mapestimation are performed. Laplacian-based phase unwrapping is thenapplied, and then background phase removal by sophisticated harmonicartifact reduction for phase data with varying spherical kernel isexecuted. Of course, the data used in the inventive concept is notlimited or limited to the data described above, and all data availablein the inventive concept may be used.

Network Architectures

FIG. 2 shows an exemplary diagram of the architecture of the cycleQSM ofthe inventive concept.

Referring to FIG. 2, the forward mapping is described by thedeterministic dipole kernel in [Equation 2], cycleQSM has only one pairof the generator and discriminator. That is, unlike the conventionalcycleGAN, in the inventive concept, a neural network may have onegenerator and one discriminator due to the dipole kernel. As a result,the training of the neural network according to the inventive conceptmay be more stable and faster than the conventional cycleGAN. To providemore information to the generator, the magnitude image is concatenatedwith the phase image. The dipole kernel for each training step isgenerated depending on the resolution and direction of the main magneticfield of each data. Accordingly, all the data set in Table 1 withdifferent spatial resolution may be fully utilized for trainingcycleQSM. Also, the restored (or reconstructed) QSM is multiplied by thebrain mask before going through the discriminator. Since, without thebrain mask, the discriminator may distinguish real QSM and fake QSM byobserving the artifact outside the brain mask region, the inventiveconcept may stabilize the training process and reduce the artifactoutside of the brain region, by multiplying the brain mask.

That is, according to the inventive concept, cycleQSM may be trainedthrough the architecture shown in FIG. 2, and when the training processis completed, only the phase image and the magnitude image may bereceived using only the actual generator to restore the QSM image.

Specifically, the upper part of cycleQSM shown in FIG. 2 starts withgenerating a quantitative susceptibility mapping image by inputting theactual phase image to the generator together with the magnitude (orscale) image. In the generated quantitative susceptibility mappingimage, the total variation (TV) loss for preserving image details andremoving noise may be calculated. Next, the phase image is reconstructedby applying the Fourier transform to the generated quantitativesusceptibility mapping image, multiplying the generated dipole kernelaccording to the resolution, and applying the inverse Fourier transformagain. A cycle-consistency loss and a gradient difference loss arecalculated by comparing the phase image reconstructed as described aboveand a phase image initially used as an input image of the generator.Furthermore, in the lower part of cycleQSM, the phase image is generatedusing the dipole kernel from the actual quantitative susceptibilitymapping image, and then the generator reconstructs the quantitativesusceptibility mapping image. Similarly, the cycle-consistency loss andthe gradient difference loss are calculated by comparing thereconstructed quantitative susceptibility mapping image with the actualquantitative susceptibility mapping image. Finally, the discriminatormay distinguish the actual quantitative susceptibility mapping imagefrom the quantitative susceptibility mapping image generated by thegenerator. In the learning phase of the generator, the least squares GAN(LSGAN) loss is minimized to deceive the discriminator, while in thelearning phase of the discriminator, the least squares adversarial lossis maximized to distinguish the real QSM from the fake QSM. Through thiscompetitive learning, the generator can reconstruct a more realisticquantitative susceptibility mapping image. In addition, the quantitativesusceptibility mapping image generated by the generator is multiplied bythe brain mask to prevent the discriminator from discriminating betweenthe real QSM and the fake QAM too easily, enabling more stable learning.

FIGS. 3A and 3B show exemplary diagrams for a generator architecture anda discriminator architecture. FIG. 3A shows an architecture of agenerator, and FIG. 3B shows an architecture of a discriminator.

As shown in FIG. 3A, the generator may use a 3D U-Net structure based onan U-Net structure. Because of the concatenation of the magnitude imageand the phase image, the generator has two input channels. The generatormay include 3×3×3 convolution, instance normalization, leaky ReLU, andnearest-neighbor upsampling layers. In addition, the skip connection viachannel concatenation is used for the generator. At the end of thenetwork, the final reconstructed QSM is generated through the 1×1×1convolution.

As shown in FIG. 3B, the discriminator may use patchGAN discriminatorand may consist of 4×4×4 convolution, instance normalization, and leakyReLU. Inputs of the discriminator are real susceptibility maps orgenerated susceptibility maps that are multiplied with brain masks.

Since there are only 8 phase images and 3 pieces of QSM label data fornetwork training, the inventive concept may use random patches duringtraining to increase the amount of data. During one epoch, the inventiveconcept may extract a total of 3000 phase and unpaired QSM randompatches of size 64×64×64, respectively. In the inference step, a phasevolume is cropped into patches of size 64×64 with a stride of 16×16×16,and all patch inference results are combined to reconstruct the entireQSM volume. Moreover, the inventive concept may apply data augmentationby flipping with respect to each axis, and rotating in a planeperpendicular to the direction of the main magnetic field.

The inventive concept may use Adam optimizer with β01=0.5, β2=0.999, andlearning rate of 0.00001 and also, choose γ=10, η=1, and ρ=0.1 for[Equation 16]. In the inventive concept, the cycleGAN is trained for 50epochs, and implemented in Python by TensorFlow.

The method of the inventive concept may be compared with severalconventional methods to verify the performance of the algorithm. First,TKD is used to replace the values in the conical surfaces of the dipolekernel with a certain threshold value, and may be expressed as in[Equation 17] below.

$\begin{matrix}{{\hat{d}\left( {\overset{\rightarrow}{k};a} \right)} = \begin{Bmatrix}{d\left( \overset{\rightarrow}{k} \right)} & {\hat{❘}{d\left( \overset{\rightarrow}{k} \right)}{❘{> a}}} \\{a \cdot {{sign}\left( {d\left( \overset{\rightarrow}{k} \right)} \right)}} & {\hat{❘}{d\left( \overset{\rightarrow}{k} \right)}{❘{\leq a}}}\end{Bmatrix}} & \left\lbrack {{Equation}17} \right\rbrack\end{matrix}$

where α is a threshold value.

Next, MEDI is also compared with our algorithm and the QSMreconstruction by MEDI can be formulated as in [Equation 18] below.

$\begin{matrix}{\min\limits_{\chi}{{{{W\left( {b - {\mathcal{F}^{- 1}\hat{d}{\mathcal{F}\chi}}} \right)}_{2}^{2}} + \lambda}}M{\nabla\chi}_{1}} & \left\lbrack {{Equation}18} \right\rbrack\end{matrix}$

where W is the structural weight matrix which is derived by themagnitude image, and M is the binary mask that contains the edgeinformation of the magnitude image.

In addition, the inventive concept may compare cycleQSM with iLSQR andALOHA-QSM and in our comparison experiments, the inventive concept mayuse a=0.1 for TKD, λ=600 for MEDI, and 30 iteration steps for iLSQR,respectively. Also, hyper-parameters for ALOHA-QSM are set toλ=10^(1.5), μ=10^(1.5) for Cornell data, and λ=10^(2.4), μ=10^(−2.2) forALOHA-QSM data, respectively.

The method according to the inventive concept may be also compared withother deep learning methods. Supervised learning using the same U-Netnetwork is compared with the method according to the inventive concept.The network in supervised learning may be trained using L1 loss during50 epochs with the learning rate of 0.0001. Next, deep image prior (DIP)is used for dipole inversion and is optimized for each volume withouttraining, and the optimization in DIP can be formulated as in [Equation19] below.

$\begin{matrix}{\min\limits_{\chi}{{{{W\left( {e^{{j\mathcal{F}}^{- 1}\hat{d}\mathcal{F}_{\chi}} - e^{jb}} \right)}_{1}} + {\lambda{{\nabla\chi}}_{1}}}}} & \left\lbrack {{Equation}19} \right\rbrack\end{matrix}$

where W is a noise weighting factor that is obtained from the magnitudeimage.

The inventive concept may use λ in [Equation 19] as 0.001 for the bestperformance. The network architecture in DIP is same as the generatoraccording to the inventive concept, but the number of channels may bereduced by half due to the GPU memory limitation. In addition, uQSM maybe used for comparison as another unsupervised learning method. uQSM mayminimize the loss in [Equation 19] with λ=0:001, but it is trained withtraining data first, then the trained network is used for thereconstruction of test data. uQSM may be trained during 50 epochs withthe learning rate of 0.0001.

To evaluate algorithms, the peak signal-to-noise ratio (PSNR) and thestructural similarity index metrics (SSIM) may be used as quantitativemetrics. PSNR and SSIM may be measured for 3D volume of Cornell datawhich has ground-truth QSM. Also, the root mean square error (RMSE)which is calculated as in [Equation 20] below may be used.

${RMSE} = \sqrt{\frac{\sum_{i = 1}^{N}\left( {\chi_{i} - {\overset{\sim}{\chi}}_{i}} \right)^{2}}{N}}$

where N is the number of pixels which in the brain mask, and x_(i) and{tilde over (X)}_(i) are pixel intensities of ground-truth andreconstructed QSM.

FIG. 4 shows an exemplary view of results of reconstructing asusceptibility mapping image using the method of the inventive conceptand various existing methods, and in FIG. 4, phase represents a phaseimage, TKD, MEDI, iLSQR, and ALOHA-QSM represent QSMs reconstructed byconventional QSM reconstruction methods, supervised represents a QSMreconstructed by a supervised learning-based deep-learning method, DIPand uQSM represent QSMs reconstructed by unsupervised learning-baseddeep learning methods, proposed represents QSM reconstructed by themethod of the inventive concept, and ground-truth represents theground-truth QSM.

As can be seen from FIG. 4, it can be seen that the QSM reconstructedfrom TKD has severe streaking artifacts, and the QSM reconstructed byMEDI does not have streaking artifacts, but provides an excessivelysmooth reconstruction result. Furthermore, the QSMs reconstructed byiLSQR and ALOHA-QSM are more realistic QSMs, but it can be seen thatstreaking artifacts remain in the reconstructed QSM,s and supervisedlearning reconstructs QSM close to the ground-truth QSM withoutstreaking artifacts. The output of DIP is smoothed and somesusceptibility values are not restored. Also, DIP requires very longreconstruction time compared to other methods because it has to beoptimized to each volume. uQSM can reconstruct QSM without label data,but uQSM generates excessively-smoothed output so that some ofstructures in the reconstructed QSM are not recognizable. On the otherhand, the method according to the inventive concept reconstructs QSMthat are close to the ground-truth data without the artifacts. Inaddition, the method according to the inventive concept requires similarreconstruction time as supervised learning.

FIGS. 5A and 5B show diagrams illustrating an example for describing ananalysis result using susceptibility values of a ground-truth image andsusceptibility values of an image reconstructed by variousreconstruction methods. FIG. 5A shows each brain structure, and FIG. 5Bshows results of linear regression analysis on the susceptibility valuesof the ground-truth image and the susceptibility values of imagesreconstructed by various reconstruction methods. Here, in FIG. 5B, thehorizontal axis represents the susceptibility values of the ground-truthimage, the vertical axis represents the susceptibility values of thereconstructed image, and the susceptibility values may be extracted fromthe gray matter structures of FIG. 5A.

As can be seen from FIGS. 5A and 5B, the deep learning-based methodsdifferent from the conventional methods show many error values,underestimation, overestimation, or the like, whereas the method of theinventive concept shows small errors and accurate linear regressionresults.

As described above, the method according to an embodiment of theinventive concept may reconstruct a quantitative susceptibility mappingimage using an unsupervised learning-based neural network generated bythe optimal transport theory.

In addition, the inventive concept provides a specific contrast imagethrough a quantitative susceptibility mapping image or providesinformation that is sensitive to biomarkers such as iron deposition andhelpful in diagnosing diseases, and thus, is applicable to all fieldsusing quantitative susceptibility mapping images, medical equipment, orthe like.

FIG. 6 illustrates a configuration of a quantitative susceptibilitymapping image processing apparatus according to an embodiment of theinventive concept, and illustrates a conceptual configuration of anapparatus for performing the methods of FIGS. 1 to 5.

Referring to FIG. 6, a quantitative susceptibility mapping imageprocessing apparatus 600 according to an embodiment of the inventiveconcept includes a receiver 610 and a reconstruction unit 620.

The receiver 610 receives a phase image and a magnitude image forreconstructing a quantitative susceptibility mapping image.

The reconstruction unit 620 reconstructs a quantitative susceptibilitymapping image corresponding to the received phase image and magnitudeimage by using an unsupervised learning-based neural network.

Here, the neural network used in the inventive concept may have a cycleGenerative Adversarial Network (cycleGAN) structure including at leastone generator and at least one discriminator to reconstruct aquantitative susceptibility mapping image.

In addition, the neural network used in the inventive concept may begenerated based on the optimal transport theory, and may be learnedusing a training dataset including non-matching data.

Furthermore, the neural network used in the inventive concept may besubjected to unsupervised learning by using a first neural network(e.g., a generator) that generates a first quantitative susceptibilitymapping image corresponding to a first phase image and a first magnitudeimage after receiving the first phase image and the first magnitudeimage, a first transform unit that performs Fourier transform on thefirst quantitative susceptibility mapping image, multiplies the firstquantitative susceptibility mapping image by a dipole kernelcorresponding to the first phase image and the first magnitude image,and performs inverse Fourier transform on the first quantitativesusceptibility mapping image to generate a second phase image, a secondtransform unit that performs Fourier transform on a second quantitativesusceptibility mapping image, which is a ground-truth image, andmultiplies the second quantitative susceptibility mapping image by adipole kernel corresponding to the second quantitative susceptibilitymapping image, and performs inverse Fourier transform on the secondquantitative susceptibility mapping image to generate a third phaseimage, a second neural network (e.g., a generator) that generates athird quantitative susceptibility mapping image corresponding to thesecond phase image and a second magnitude image after receiving thesecond phase image and the second magnitude image, and a third neuralnetwork (e.g., a discriminator) that discriminates the firstquantitative susceptibility mapping image and the second quantitativesusceptibility mapping image. Here, the first neural network and thesecond neural network may be the same neural network.. Here, the firstneural network and the second neural network may be the same neuralnetwork.

Furthermore, the neural network may be subjected to unsupervisedlearning based on a cycle-consistency loss and a gradient differenceloss calculated by comparing the first phase image with the second phaseimage, or the second quantitative susceptibility mapping image with thethird quantitative susceptibility mapping image and a total variationloss for the first quantitative susceptibility mapping image, and anadversarial loss between the first quantitative susceptibility mappingimage and the second quantitative susceptibility mapping image.

The third neural network may distinguish a quantitative susceptibilitymapping image obtained by multiplying the first quantitativesusceptibility mapping image by a preset mask and the secondquantitative susceptibility mapping image.

The neural network used in the inventive concept may include any one ofa neural network based on a convolution framelet and a neural networkincluding a pooling layer and an unpooling layer.

Although the description is omitted with reference to the apparatus ofFIG. 6, components constituting FIG. 6 may include all the contentsdescribed with reference to FIGS. 1 to 5, which are obvious to thoseskilled in the art.

The apparatus described herein may be implemented with hardwarecomponents and software components and/or a combination of the hardwarecomponents and the software components. For example, the apparatus andcomponents described in the embodiments may be implemented using one ormore general-purpose or special purpose computers, such as, for example,a processor, a controller and an arithmetic logic unit (ALU), a digitalsignal processor, a microcomputer, a field programmable array (FPA), aprogrammable logic unit (PLU), a microprocessor or any other devicecapable of executing and responding to instructions. The processingdevice may run an operating system (OS) and one or more softwareapplications that run on the OS. The processing device also may access,store, manipulate, process, and create data in response to execution ofthe software. For convenience of understanding, one processing device isdescribed as being used, but those skilled in the art will appreciatethat the processing device includes a plurality of processing elementsand/or multiple types of processing elements. For example, theprocessing device may include multiple processors or a single processorand a single controller. In addition, different processingconfigurations are possible, such a parallel processors.

The software may include a computer program, a piece of code, aninstruction, or some combination thereof, for independently orcollectively instructing or configuring the processing device to operateas desired. Software and/or data may be embodied in any type of machine,component, physical or virtual equipment, computer storage medium ordevice that is capable of providing instructions or data to or beinginterpreted by the processing device. The software also may bedistributed over network coupled computer systems so that the softwareis stored and executed in a distributed fashion. In particular, thesoftware and data may be stored by one or more computer readablerecording mediums.

The above-described methods may be embodied in the form of programinstructions that can be executed by various computer means and recordedon a computer-readable medium. The computer readable medium may includeprogram instructions, data files, data structures, and the like, aloneor in combination. Program instructions recorded on the media may bethose specially designed and constructed for the purposes of theinventive concept, or they may be of the kind well-known and availableto those having skill in the computer software arts. Examples ofcomputer readable recording media include magnetic media such as harddisks, floppy disks and magnetic tape, optical media such as CD-ROMs,DVDs, and magnetic disks such as floppy disks, Magneto-optical media,and hardware devices specifically configured to store and executeprogram instructions, such as ROM, RAM, flash memory, and the like.Examples of program instructions include not only machine code generatedby a compiler, but also high-level language code that can be executed bya computer using an interpreter or the like.

Although the embodiments have been described by the limited embodimentsand the drawings as described above, various modifications andvariations are possible to those skilled in the art from the abovedescription. For example, the described techniques may be performed in adifferent order than the described method, and/or components of thedescribed systems, structures, devices, circuits, etc. may be combinedor combined in a different form than the described method, or othercomponents, or even when replaced or substituted by equivalents, anappropriate result can be achieved.

Therefore, other implementations, other embodiments, and equivalents tothe claims are within the scope of the following claims.

According to the embodiments of the inventive concept, it is possible toreconstruct a quantitative susceptibility mapping image using anunsupervised learning-based neural network.

Furthermore, since the quantitative susceptibility mapping imageprovides specific image contrast unlike magnetic resonance imaging (MRI)or provides information that is sensitive to biomarkers such as irondeposition and helpful in diagnosing diseases, the inventive concept isapplicable to all fields using quantitative susceptibility mappingimages, medical equipment, or the like.

While the inventive concept has been described with reference toexemplary embodiments, it will be apparent to those skilled in the artthat various changes and modifications may be made without departingfrom the spirit and scope of the inventive concept. Therefore, it shouldbe understood that the above embodiments are not limiting, butillustrative.

What is claimed is:
 1. A quantitative susceptibility mapping imageprocessing method for reconstructing a quantitative susceptibilitymapping (QSM) image, the method comprising: receiving a phase image anda magnitude image for reconstructing the quantitative susceptibilitymapping image; and reconstructing the quantitative susceptibilitymapping image corresponding to the received phase image and the receivedmagnitude image using an unsupervised learning-based neural network,wherein the neural network includes a cycle-consistency generativeadversarial network (cycleGAN) structure including at least onegenerator and at least one discriminator to reconstruct the quantitativesusceptibility mapping image.
 2. The quantitative susceptibility mappingimage processing method of claim 1, wherein the neural network isgenerated based on an optimal transport theory.
 3. The quantitativesusceptibility mapping image processing method of claim 1, wherein theneural network is trained using a training dataset containingnon-matching data.
 4. The quantitative susceptibility mapping imageprocessing method of claim 1, wherein the neural network is subjected tounsupervised learning by using: a first neural network configured togenerate a first quantitative susceptibility mapping image correspondingto a first phase image and a first magnitude image after receiving thefirst phase image and the first magnitude image; a first transform unitconfigured to perform Fourier transform on the first quantitativesusceptibility mapping image, multiply the first quantitativesusceptibility mapping image by a dipole kernel corresponding to thefirst phase image and the first magnitude image, and perform inverseFourier transform on the first quantitative susceptibility mapping imageto generate a second phase image; a second transform unit configured toperform Fourier transform on a second quantitative susceptibilitymapping image, which is a ground-truth image, and multiply the secondquantitative susceptibility mapping image by a dipole kernelcorresponding to the second quantitative susceptibility mapping image,and perform inverse Fourier transform on the second quantitativesusceptibility mapping image to generate a third phase image; a secondneural network configured to generate a third quantitativesusceptibility mapping image corresponding to the second phase image anda second magnitude image after receiving the second phase image and thesecond magnitude image; and a third neural network configured todiscriminate the first quantitative susceptibility mapping image and thesecond quantitative susceptibility mapping image.
 5. The quantitativesusceptibility mapping image processing method of claim 4, wherein theneural network is subjected to the unsupervised learning based on acycle-consistency loss and a gradient difference loss calculated bycomparing the first phase image with the second phase image, or thesecond quantitative susceptibility mapping image with the thirdquantitative susceptibility mapping image and a total variation loss forthe first quantitative susceptibility mapping image, and an adversarialloss between the first quantitative susceptibility mapping image and thesecond quantitative susceptibility mapping image.
 6. The quantitativesusceptibility mapping image processing method of claim 4, wherein thethird neural network distinguishes a quantitative susceptibility mappingimage obtained by multiplying the first quantitative susceptibilitymapping image by a preset mask and the second quantitativesusceptibility mapping image.
 7. The quantitative susceptibility mappingimage processing method of claim 1, wherein the neural network includesany one of a neural network based on a convolution framelet and a neuralnetwork including a pooling layer and an unpooling layer.
 8. Aquantitative susceptibility mapping image processing method forreconstructing a quantitative susceptibility mapping (QSM) image, themethod comprising: receiving a phase image and a magnitude image forreconstructing the quantitative susceptibility mapping image; andreconstructing the quantitative susceptibility mapping imagecorresponding to the received phase image and the received magnitudeimage using an unsupervised learning-based neural network generatedbased on an optimal transport theory, wherein the neural networkincludes a cycle-consistency generative adversarial network (cycleGAN)structure including at least one generator and at least onediscriminator to reconstruct the quantitative susceptibility mappingimage.
 9. A quantitative susceptibility mapping image processingapparatus for reconstructing a quantitative susceptibility mapping (QSM)image, the apparatus comprising: a receiver configured to receive aphase image and a magnitude image for reconstructing the quantitativesusceptibility mapping image; and a reconstruction unit configured toreconstruct the quantitative susceptibility mapping image correspondingto the received phase image and the received magnitude image using anunsupervised learning-based neural network, wherein the neural networkincludes a cycle-consistency generative adversarial network (cycleGAN)structure including at least one generator and at least onediscriminator to reconstruct the quantitative susceptibility mappingimage.
 10. The quantitative susceptibility mapping image processingapparatus of claim 9, wherein the neural network is generated based onan optimal transport theory.
 11. The quantitative susceptibility mappingimage processing apparatus of claim 9, wherein the neural network istrained using a training dataset containing non-matching data.
 12. Thequantitative susceptibility mapping image processing apparatus of claim9, wherein the neural network is subjected to unsupervised learning byusing: a first neural network configured to generate a firstquantitative susceptibility mapping image corresponding to a first phaseimage and a first magnitude image after receiving the first phase imageand the first magnitude image; a first transform unit configured toperform Fourier transform on the first quantitative susceptibilitymapping image, multiply the first quantitative susceptibility mappingimage by a dipole kernel corresponding to the first phase image and thefirst magnitude image, and perform inverse Fourier transform on thefirst quantitative susceptibility mapping image to generate a secondphase image; a second transform unit configured to perform Fouriertransform on a second quantitative susceptibility mapping image, whichis a ground-truth image, and multiply the second quantitativesusceptibility mapping image by a dipole kernel corresponding to thesecond quantitative susceptibility mapping image, and perform inverseFourier transform on the second quantitative susceptibility mappingimage to generate a third phase image; a second neural networkconfigured to generate a third quantitative susceptibility mapping imagecorresponding to the second phase image and a second magnitude imageafter receiving the second phase image and the second magnitude image;and a third neural network configured to discriminate the firstquantitative susceptibility mapping image and the second quantitativesusceptibility mapping image.
 13. The quantitative susceptibilitymapping image processing apparatus of claim 12, wherein the neuralnetwork is subjected to the unsupervised learning based on acycle-consistency loss and a gradient difference loss calculated bycomparing the first phase image with the second phase image, or thesecond quantitative susceptibility mapping image with the thirdquantitative susceptibility mapping image and a total variation loss forthe first quantitative susceptibility mapping image, and an adversarialloss between the first quantitative susceptibility mapping image and thesecond quantitative susceptibility mapping image.
 14. The quantitativesusceptibility mapping image processing apparatus of claim 12, whereinthe third neural network distinguishes a quantitative susceptibilitymapping image obtained by multiplying the first quantitativesusceptibility mapping image by a preset mask and the secondquantitative susceptibility mapping image.
 15. The quantitativesusceptibility mapping image processing apparatus of claim 9, whereinthe neural network includes any one of a neural network based on aconvolution framelet and a neural network including a pooling layer andan unpooling layer.